This lecture series was given in the Autumn Term, 2006 at the Wellcome Trust Centre for Human Neuroimaging at UCL. It covers a number of mathematical methods that are used in the analysis of brain imaging data. Each lecture describes a different category of model and shows how it is applied to a particular aspect of brain imaging analysis. The applications cover data from functional Magnetic Resonance Imaging (fMRI), Magnetoencephalography (MEG) and Electroencephalography (EEG).
Download all course notes: PDF
1. General Linear Models I
- Maximum likelihood estimation
- Regression and correlation
- Linear algebra
- Functions of random vectors
- Multiple regression and partial correlation
- Application: fMRI time series
Lecture notes and code can be downloaded from this archive: ZIP
2. General Linear Models II
- Estimating error variance
- Comparing nested models
- Transforming probability densities
- Contrasts
- Hemodynamic basis functions
- Application: fMRI time series
Notes and matlab code can be downloaded here: ZIP
- Gaussian processes
- Covariance functions
- Upcrossings of one-dimensional processes
- Euler characteristic
- Application: Detecting activations in fMRI data
Notes and matlab code for 1D/2D fields can be downloaded here: ZIP
- More Linear Algebra
- Principal component analysis
- Singular Value Decomposition
- Structural Equation Modelling
- Granger causality
- Application: PET & fMRI connectivity analyses
Notes and matlab code can be downloaded here: ZIP
- GLMs with arbitrary error covariance
- Weighted Least Squares
- Restricted Maximum Likelihood
- Application: fMRI time series analysis with correlated errors
- Hierarchical Models
- Application: Analysis of imaging data from a group
Notes and matlab code for ReML estimation of variance components can be downloaded here: ZIP
- Bayes rule for Gaussians
- Bayesian GLMs
- Parametric Empirical Bayes (PEB)
- Expectation Maximisation (EM)
- Application: EEG source reconstruction
Notes and matlab code for EM example can be downloaded here: ZIP
- Bayes factors and odds ratios
- Model evidence for Bayesian GLMs
- Accuracy and complexity (AIC/BIC)
- Bias-variance decomposition
- Application: EEG source reconstruction
- Bayesian model averaging
- Application: Nonlinear EEG source reconstruction
Notes can be downloaded here: PDF
- Fourier series and periodograms
- Autocorrelation and power spectral density
- Cross-correlation and cross spectral density
- Coherence and Phase
- Welch and multitaper methods
- Localisation of MEG Gamma activity
Notes and matlab code for sunspot spectra can be downloaded here: ZIP
9. Approximate Bayesian Inference
- Laplace approximation
- Kullback-Liebler divergence
- Variational Bayes and EM
- Mixture models
- Application: Group analysis of imaging data
Notes can be downloaded here: PDF
- Central Limit Theorem
- Independent Component Analysis
- Application: EEG artifact removal
- Discriminant Analysis
- Application: Estimating perceptual state from fMRI
Notes can be downloaded here: PDF